Optimal Bank Capital

David MilesMonetary Policy Committee

The Bank of EnglandJune 2011

5

10

15

20

25

30

35

40

-3%

-2%

-1%

0%

1%

2%

3%

4%

5%

1880 1900 1920 1940 1960 1980 2000

Real GDP growth 10y-MA

Leverage (rhs)(a)

(c)(b)

UK Banks’ leverage and real GDP growth (10-year moving average)

Source: United Kingdom: Sheppard, D (1971), The growth and role of UK financial institutions 1880-1962, Methuen, London; Billings, M and Capie, F (2007), 'Capital in British banking', 1920-1970, Business History, Vol 49(2), pages 139-162; BBA, ONS published accounts and Bank calculations. (a) UK data on leverage use total assets over equity and reserves on a time-varying sample of banks, representing the majority of the UK banking system, in terms of assets. Prior to 1970 published accounts understated the true level of banks' capital because they did not include hidden reserves. The solid line adjusts for this. 2009 observation is from H1. (b) Change in UK accounting standards. (c) International Financial Reporting Standards (IFRS) were adopted for the end-2005 accounts. The end-2004 accounts were also restated on an IFRS basis. The switch from UK GAAP to IFRS reduced the capital ratio of the UK banks in the sample by approximately 1 percentage point in 2004.

19201923

19261929

19321935

19381941

19441947

19501953

19561959

19621965

19681971

19741977

19801983

19861989

0

1

2

3

4

5

6

0

2

4

6

8

10

12

14

16

18

20Spread Spread - 5yr MA Leverage (rhs)%

Leverage and spreads of average business loan rates charged by US commercial banks over 3-month Treasury bills

Source: -Business loan spreads from Homer and Sylla (1991). The spread is calculated as the difference between the average business loan rate and the average 3-month Treasury bill rate.-Leverage from Berger, A, Herring, R and SzegÖ, G (1995), 'The role of capital in financial institutions', Journal of banking and finance, Vol 19, pp 393-430. Leverage is defined as the ratio of assets to the book value of equity.

Assuming that debt is riskless,

The link between equity beta and the leverage ratio: Theory

ED

D

ED

Edebtequityassets

assetsequity E

ED

Average equity beta across major UK banks, 1997-2010

i indices banks and t time periodsX includes leverage, year dummies

Estimation techniques: OLS, fixed effects, and random effects. Year effects included.

The link between equity beta and the leverage ratio: Estimation

titiiti bX ,',,

ˆ

Bank equity beta and leverage: Estimation results

Variable OLS FE RE

leverage 0.025 0.031 0.025

(4.22) (3.49) (5.35)

Const 1.238 1.072 1.237

(3.99) (3.72) (5.55)

R-sqr _overall 0.671 0.664 0.671

R-sqr _between 0.634 0.670

R-sqr _within 0.658 0.654

Regression of bank equity beta on leverage, measured as total assets/tier 1 capital. All specifications include year effects. In all three regressions, standard errors are robust to clustering effects at the bank level. Coefficient t statistics are in parenthesis. A Hausman test is used to compare FE and RE estimators. The null hypothesis is that the differences in coefficients are not systemic. Chi-square (12) = 2.84 with P-value = 0.99.

The link between the required return on equity and leverage

mRiskpremiuRR equityfequity

The CAPM states that the required return on equity can be expressed as

Inserting our estimate of the link between beta and the leverage ratio yields

mRiskpremiuE

EDbaRR fequity

ˆˆ

The link between the required return on equity and leverage: Results

At a leverage ratio of (D+E)/E = 30,

%85.14

%530*03.007.1%5

ˆˆ

mRiskpremiuleveragebaRR fequity

%3.5

30/29%530/1%85.14

ED

DR

ED

ERWACC fequity

If leverage fell to 15, the required return on equity would fall to 12.6%.; the WACC would rise to 5.5% - a rise of about 20bp. The MM offset is about one half of its “theoretical” level

OLS FE RE

leverage 0.602 0.692 0.602

(6.58) (3.76) (6.81)

_cons -1.405 -1.693 -1.405

(-4.45) (-2.69) (-4.35)

R-sqr _overall 0.62 0.66 0.67

R-sqr _between 0.54 0.61

R-sqr _within 0.64 0.636

Bank equity beta and leverage: Estimation results(log specification)

Regression of the log of bank equity beta on log leverage, measured as total assets/tier 1 capital. All specifications include year effects. In all three regressions, standard errors are robust to clustering effects at the bank level. Coefficient t statistics are in parenthesis.

Variable OLS FE RE

leverage 0.0021 0.0023 0.0023

(2.52) (1.97) (2.52)

cons 0.0520 0.0467 0.0456

(1.59) (1.45) (1.59)

R-sqr _overall 0.0801 0.0801 0.0801

R-sqr _between 0.2037 0.2037

R-sqr _within 0.0584 0.0584

Required return on capital and leverage: Estimation results

Regression of banks’ required return on equity (E/P) on leverage. In all three regressions, standard errors are robust to clustering effect at the bank level.

Translating changes in bank funding costs into changes in output

• Funding cost increase passed on to customers• Households’ and non-financial firms’ cost of capital increases• Reduction in investment and output: Estimation using a CES

production function

1|rY

• ε = elasticity of output with respect to funding cost;• σ = elasticity of substitution between capital and labour;• α = elasticity of output with respect to capital (capital share).

Tax effect, no M-M

Tax effect, 45% M-M

Base case: no tax effect, 45%

M-M

No tax effect and 75% M-M

Change in banks WACC 38.0 22.5 17.9 7.7

Change in PNFC WACC 12.7 7.5 6.0 2.6

Fall in long run GDP 31.7 18.8 14.9 6.4

Present value of GDP lost 1268 751 596 256

Economic Impact of halving Leverage from 30 to 15 relative to Tier 1 capital; or from 50 to 25 based on assets to CET1 – Basis Points

• Equivalent to doubling capital as percent of risk-weighted assets.

Base case (no tax effect & 45% MM)

Higher discount rate

(@ 5%)

Lower share of banks in

PNFC finance (@ 16%)

Higher Equity Risk Premium (@

7.5%)

Change in banks WACC 17.9 17.9 17.9 26.8

Change in PNFC WACC 6.0 6.0 2.9 8.9

Fall in long run GDP 14.9 14.9 7.1 22.3

Present value of GDP lost 596 298 286 894

Sensitivity of base case estimates to changes in various assumptions – basis points

Quantifying the benefits: Frequency distribution of annual falls in GDP

Annual GDP fall >20% >15% >10% >5% >2% >0%

Observed frequency (%) 0.40 1.21 2.48 6.95 13.8 27.10

Frequency implied by normal distribution (%)

0.006 0.16 1.90 11.58 25.17 37.50

Quantifying the benefits: The distribution of permanent falls in GDP

• Assume that mean per-capital GDP, y, follows a random walk with drift and two random components:

111loglog tttt vuyy

where the random components follow

/2y probabilit with

/2y probabilit with

y probabilit with

1y probabilit with 0

qc

qc

pb

qp

vt

Annual GDP Growth: Comparing the Economic Model with Actual Data (1821-2008)

Parameter Value

Std. deviation of GDP growth in normal times ( ) 3.1%

Average productivity growth ( ) 2.1%

Annual probability of extreme negative shock (p) 0.7%

Scale of extreme negative shock (-b) -35%

Annual probability of less extreme, symmetric shock (q) 7.0%

Scale of less extreme, symmetric shock (c) ±12.5%

Quantifying the benefits: Key parameters

Actual data Model Prediction

Mean (%) 1.81 1.80

Standard deviation (%) 5.7 5.9

Skewness -2.40 -2.65

Excess Kurtosis 39.0 20.0

observations 4472

Percent of observations less than...

-20% 0.4 0.7

-15% 1.2 1.1

-10% 2.5 2.9

-5% 7.0 5.0

-2% 13.8 12.7

0% 27.1 27.1

Percent of observations more than...

0% 72.8 72.9

+2% 51.1 50.9

+5% 19.5 19.5

+10% 3.6 3.7

+15% 1.3 1.9

+20% 0.4 0.2

Actual growth in GDP per capita (1821-2008) and model fitted to GDP growth: detail

Finding the optimal capital ratio: Expected costs of financial crises and macro cost of banks capital

Net benefit of capital assuming financial crises have some permanent effect on GDP growth

Net benefit of capital assuming financial crises have no permanent effect on GDP growth

Optimal capital ratios considering full distribution of bad events

Crises have some permanent effects

on GDP growthCrises have no permanent effects

on GDP growth

Base cost of capital 19% 17%

Lower cost of capital 47% 18%

Higher cost of capital 18% 16%

Optimal capital ratios ignoring the most extreme bad events

Crises have some permanent effects

on GDP growthCrises have no permanent effects

on GDP growth

Base cost of capital 19% 17%

Lower cost capital 20% 18%

Higher cost capital 18% 16%

Backup slides

• Detailed estimation results• Actual growth in GDP per capita (1821-2008)

and model fitted to GDP growth

Bank equity beta and leverage: Estimation results (details)

Variable OLS FE RE

leverage 0.025 0.031 0.025

(4.22) (3.49) (5.35)

Const 1.238 1.072 1.237

(3.99) (3.72) (5.55)

R-sqr

R-sqr _overall 0.671 0.664 0.671

R-sqr _between 0.634 0.670

R-sqr _within 0.658 0.654

F-test or Wald test 13.3 7.54 122

Prob>F 0.00 0.00 0.00

Year effect yes yes yes

Regression of bank equity beta on leverage, measured as total assets/tier 1 capital. All specifications include year effects. In all three regressions, standard errors are robust to clustering effects at the bank level. Coefficient t statistics are in parenthesis. A Hausman test is used to compare FE and RE estimators. The null hypothesis is that the differences in coefficients are not systemic. Chi-square (12) = 2.84 with P-value = 0.99.

Variable OLS FE RE

leverage 0.0021 0.0023 0.0023

(2.52) (1.97) (2.52)

cons 0.0520 0.0467 0.0456

(1.59) (1.45) (1.59)

r-square 0.0801

r-sq_overall 0.0801 0.0801

r-sq_between 0.2037 0.2037

r-sq_within 0.0584 0.0584

F-test or Wald test 4.1781 3.8941 6.35*

Prob > F 0.05 0.05 0.01

Required return on capital and leverage (detail)

*In the random effect regression, this is the Wald test statistics for overall significance of the repressors

Regression of banks’ required return on equity (E/P) on leverage. In all three regressions, standard errors are robust to clustering effect at the bank level.